Disturbance rejection control for vibration suppression of piezoelectric laminated thin-walled structures

Journal of Sound and Vibration 333 (2014) 1209–1223

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Disturbance rejection control for vibration suppression of piezoelectric laminated thin-walled structures S.Q. Zhang a,n, H.N. Li a, R. Schmidt a, P.C. Müller b a b

Institute of General Mechanics, RWTH Aachen University, Templergraben 64, D-52062 Aachen, Germany Safety Control Engineering, University of Wuppertal, Gauss-Str. 20, 42097 Wuppertal, Germany

a r t i c l e i n f o

abstract

Article history: Received 27 May 2013 Received in revised form 27 September 2013 Accepted 21 October 2013 Handling Editor: D.J. Wagg Available online 22 November 2013

Thin-walled piezoelectric integrated smart structures are easily excited to vibrate by unknown disturbances. In order to design and simulate a control strategy, firstly, an electro-mechanically coupled dynamic finite element (FE) model of smart structures is developed based on first-order shear deformation (FOSD) hypothesis. Linear piezoelectric constitutive equations and the assumption of constant electric field through the thickness are considered. Based on the dynamic FE model, a disturbance rejection (DR) control with proportional-integral (PI) observer using step functions as the fictitious model of disturbances is developed for vibration suppression of smart structures. In order to achieve a better dynamic behavior of the fictitious model of disturbances, the PI observer is extended to generalized proportional-integral (GPI) observer, in which sine or polynomial functions can be used to represent disturbances resulting in better dynamics. Therefore the disturbances can be estimated either by PI or GPI observer, and then the estimated signals are fed back to the controller. The DR control is validated by various kinds of unknown disturbances, and compared with linear-quadratic regulator (LQR) control. The results illustrate that the vibrations are better suppressed by the proposed DR control. & 2013 Elsevier Ltd. All rights reserved.

1. Introduction Thin-walled structures with integrated smart materials, which are so-called smart structures, are increasingly applied in many fields of technology for vibration control and noise control. Because of high costs of experimental investigations for vibration control, simulations based on both refined structural modeling and control design are necessary and important. In order to predict the dynamic behavior of piezoelectric integrated smart structures, an efficient and accurate electromechanically coupled model is needed. There are many theories available in the literature, based on which three- and twodimensional FE methods have been developed for piezolaminated smart structures. Dube et al. [1], Kapuria and Dube [2], Ray et al. [3], Tzou and Tseng [4], Sze and Yao [5], etc., proposed three-dimensional piezoelectric coupled FE models. In contrast to two-dimensional FE methods, three-dimensional ones provide more precise models, but with large size of the models that lead to high computation time. For thin-walled smart structures, two-dimensional FE methods based on various hypotheses are mostly employed, due to relatively high accuracy of the models and less computation time. Twodimensional FE methods based on Kirchhoff–Love plate/shell theory (known as classical plate/shell theory) were developed

n

Corresponding author. Tel.: þ49 241 8098286; fax: þ 49 241 8092231. E-mail addresses: [email protected], [email protected] (S.Q. Zhang).

0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.10.024

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by [6–10] among many others. Comparing with Kirchhoff–Love shell theory, Reissner–Mindlin or FOSD plate/shell theory, see e.g. [11,12], considers additionally the transverse shear strains. FE methods based on FOSD theory for piezoelectric integrated smart structures can be found, e.g. in [13–18]. Two-dimensional theories based on third- or higher-order shear deformation hypotheses were developed by Reddy [19], Hanna and Leissa [20] among many others, and applied to FE models for smart structures, e.g. by Correia et al. [21], Loja et al. [22], Soares et al. [23], Correia [24], Schmidt and Vu [25], which have a more precise description of strains along the thickness direction compared with FOSD theory. Additionally, some other shear deformation hypotheses, like, e.g. zigzag [26,27] and layerwise [28] shear deformation theories, can be found in the literature. The majority of papers that are available developed FE models by linear plate or shell theories. However, geometrically linear theories are only valid in the range of small deformations. When structures undergo large deformations, nonlinear theories have to be considered. Most of papers that presented nonlinear FE models are using von Kármán type nonlinear theories based on various hypotheses, e.g. classical plate theory [29–31], FOSD hypothesis [32–35], third- and higher-order shear deformation hypotheses [25,36,37]. Additionally, nonlinear theories taking into account moderate or unrestricted finite rotations have been used for FE modeling of piezolaminated smart structures by Schmidt and Lentzen [38], Chróscielewski et al. [39,40], Zhang and Schmidt [41]. Yi et al. [42] developed a three-dimensional fully geometrically nonlinear FE model for transient analysis of piezolaminated pates and shells. Models of piezoelectric integrated smart structures are necessary for control design. Moreover, the control strategy determines the dynamic behavior of smart structures. Therefore, vibrations can be significantly suppressed by a well designed controller. One of the most popular control schemes for vibration suppression of smart structures is negative velocity proportional feedback control based on the models that are derived by using various theories, which can be found in [8,14,43–55] among many others. Many papers can be found in the literature which implemented LQR control using the FE models based on, e.g. classical theory [56,57], Timoshenko beam theory [43], FOSD hypothesis [14], three-dimensional FE method [58] and others [59]. Since LQR control is a full state feedback control, all the state variables have to be measured in this method, such that LQR control cannot be implemented into real systems in most of the cases. In light of this shortcoming, linear quadratic Gaussian (LQG) control was implemented by Vasques and Rodrigues [47], which estimates the state variables using the measured signals. Additionally, Stavroulakis et al. [60] implemented LQR control and robust H2 control with the model based on Euler–Bernoulli beam theory, and compared the results with each other. Apart from negative velocity proportional feedback control and optimal control, many other control laws, e.g. Lyapunov feedback control [14,43], bang-bang control [61], robust control [62], are implemented based on the finite element models. An independent modal space control has been implemented by Chen and Shen [63], Lin and Nien [64] and Bhattacharya et al. [65] using multipiezoelectric sensors and actuators. Valliappan and Qi [58] designed a prediction control algorithm for vibration control of a piezoelectric patch bonded beam. Moreover, an intelligent control, fuzzy logic control, has been developed by Shirazi et al. [66] for vibration suppression of functionally graded rectangular plate bonded with piezoelectric patches by using classical plate theory. The proposed control strategies cited above were simulated based on geometrically linear theories. Additionally, there are only a few papers in the literature considering geometrically nonlinear theory in vibration control simulation but using very simple control strategies like negative velocity proportional feedback control, see [35,67]. A literature review reveals that the majority of papers considered only simple control strategies for vibration suppression of smart structures. Additionally, most of the proposed control schemes mentioned above did not take disturbances into account as state variables fed back to the controller, since these disturbances are unknown and unmeasurable. However, they are the main causes of the vibrations. In control engineering, several methods have been proposed for the estimation of unknown disturbances, e.g. full- and reduced-order observer [68,69], PI observer [70–73], and sliding-mode observer [74,75]. Nevertheless, only a few of them compensate the estimated disturbance into the closed-loop system to improve the control effects. The aim of this paper, firstly, is to develop an electro-mechanically coupled dynamic FE model based on the FOSD hypothesis. In the model, an eight-node quadrilateral element with five mechanical degrees of freedom per node and one electrical degree of freedom for each piezoelectric layer is employed. Based on the dynamic model, a disturbance rejection control with a PI or GPI observer is developed, the former one of which was earlier proposed and developed by Müller [76,77] and applied to various nonlinear problems by Müller [78], Söffker and Müller [70,73]. In such a case, the unknown disturbances can be nonlinearly estimated by PI or GPI observer and compensated in the closed-loop control system for vibration suppression of piezolaminated smart structures. The control strategy is implemented into a cantilevered piezolaminated smart beam under various kinds of disturbances. 2. Finite element model 2.1. Strain–displacement relations The electro-mechanically coupled FE model of piezolaminated smart structures is developed based on the FOSD hypothesis. For FOSD plate and shell theory, the covariant components, v1, v2 and v3, of the displacement vector u for an arbitrary point in the shell space can be expressed by five generalized displacements as 0

1

vα ðΘ1 ; Θ2 ; Θ3 Þ ¼ vα ðΘ1 ; Θ2 Þ þ Θ3 vα ðΘ1 ; Θ2 Þ;

α ¼ 1; 2

(1)

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Fig. 1. Degrees of freedom at arbitrary mid-surface point.

0

v3 ðΘ1 ; Θ2 ; Θ3 Þ ¼ v3 ðΘ1 ; Θ2 Þ:

(2)

Here, Θ1, Θ2, and Θ3 represent the axes of the curvilinear coordinate system, which can be Cartesian, cylindrical, spherical 0

0

0

or any other coordinates, with the thickness direction defined as Θ3-line. Furthermore, v1 , v2 and v3 are the covariant 0

1

1

components of the mid-surface translational displacement vector u, and v1 and v2 the covariant components of the 1

generalized rotational displacement vector u ¼ a 3 n. Here, n is the unit normal vector of the undeformed configuration and a 3 denotes the covariant base vector in direction of the parameter line Θ3 in the deformed configuration. For linear 1

1

theory, v1 and v2 can be shown to be the mid-surface rotations about the Θ2 and Θ1 axes, respectively (see Fig. 1). According to the FOSD hypothesis, the Green–Lagrange strain tensor components for the in-plane, transverse shear and transverse normal terms, respectively, can be obtained as 0

1

2

ɛ αβ ¼ ɛαβ þ Θ3 ɛαβ þ ðΘ3 Þ2 ɛαβ 0

(3)

ɛα3 ¼ ɛ α3

(4)

ɛ33 ¼ 0:

(5)

Here and in the following Greek indices range from 1 to 2. Since no thickness change is assumed, the transverse normal strain is treated as zero, shown in Eq. (5). The components of the Green–Lagrange strain tensors are given by 0

0

0

λ0

1

2ɛαβ ¼ φαβ þ φβα 1

1

(6) δ0

2ɛαβ ¼ vαjβ  bβ φλα þvβjα  bα φδβ λ1

2

δ1

2ɛαβ ¼ bβ vλjα  bα vδjβ 1

0

0

δ0

2ɛ α3 ¼ vα þ v3;α þ bα vδ

(7) (8) (9)

with 0

0

0

n

n

φλα ¼ vλjα  bλα v3 n

vλjα ¼ vλ;α Γ δλα vδ : λ

Here, bλα and bα are the covariant and mixed components of the curvature tensor, Γ δλα the Christoffel symbols of the second n n kind, and vλjα the covariant derivatives of vλ with respect to Θα , in which the indices n over head vary between 0 and 1. 2.2. Constitutive equations The piezoelectric materials have converse and direct effects and can be used as both actuators and sensors. Due to the assumption of small strains and week electric field, linear constitutive equations of piezoelectric material are employed in the present method. They are given in matrix form as r ¼ cε  eT E

(10)

D ¼ eε þ ϵE

(11)

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where r, ε, D, and E denote the stress vector, the strain vector, the electric displacement vector, and the electric field vector, respectively. Here, the electric field intensity is assumed to be constant through the thickness of the piezoelectric layer, and equal to the negative gradient of the electric potential ϕ, which is expressed as E ¼  grad ϕ ¼ Bϕ ϕ

(12)

where Bϕ is the electric field matrix. In Eqs. (10) and (11), c denotes the elasticity constant matrix, e ¼ dc, d is the piezoelectric constant matrix, and ϵ is the dielectric constant matrix. 2.3. Dynamic model By applying the FE method and Hamilton's principle, one obtains the electro-mechanically coupled dynamic FE model, namely the equation of motion and the sensor equation, respectively, as Muu q€ þ Cuu q_ þKuu q þ Kuϕ ϕa ¼ Fue

(13)

Kϕu qþ Kϕϕ ϕs ¼ Gϕe

(14)

in which Muu , Cuu , Kuu , Kuϕ , Kϕu , and Kϕϕ denote the mass, the damping, the stiffness, the piezoelectric coupled stiffness, the coupled capacity, and the piezoelectric capacity matrices, respectively. Furthermore, Fue is the vector of external force, and Gϕe the external electric charge vector. In Eqs. (13) and (14), the vector q contains all mechanical degrees of freedom at the discretized nodes, ϕa the voltages that are applied on the actuators, and ϕs the sensor output voltages. The matrices involved in Eqs. (13) and (14) are calculated by Z (15) Muu ¼ ρNTv ZTu Zu Nv dV V

Z Kuu ¼

BTu cBu dV

V

(16)

Z Kuϕ ¼ KTϕu ¼ 

V

BTu eT Bϕ dV

(17)

Z Kϕϕ ¼  Z Fue ¼

V

BTϕ ϵBϕ dV

V

(18)

Z NTv ZTu f b dV þ

Ω

NTv ZTu f s dΩþ NTv ZTu f c

(19)

Z Gϕe ¼ 

Ω

ϱ dΩ  Q c

(20)

where ρ is the density, Nv the shape function matrix, and Bu the strain field matrix. Furthermore, f b , f s and f c are the vectors of body, surface and concentrated force, respectively, ϱ and Q c denote the surface and concentrated electric charge vectors, respectively. The damping matrix Cuu is calculated by Rayleigh damping coefficients computation method, which is linear with respect to the mass and stiffness matrices. 2.4. Model decomposition and reduction A truncated modal matrix [79] Sr , which includes the first r modes, is introduced for decomposition and reduction of the modes of smart structure as q ¼ Sr zr :

(21)

Here, zr denotes the reduced modal coordinates. The decomposed and reduced equation of motion can be achieved by substituting Eq. (21) into Eq. (13) and left-multiplying by the transposed modal matrix as ~ uu z€ r þ C~ uu z_ r þ K ~ uu zr ¼ ST Fue  ST Kuϕ ϕ M a r r

(22)

~ uu are the modal mass, damping and stiffness matrices, respectively, which are diagonal. Further we ~ uu , C~ uu and K where M assume that no extra external electric charges are applied on the piezoelectric patch acting as sensor, which means Gϕe ¼ 0, such that Eq. (14) becomes Kϕu Sr zr þKϕϕ ϕs ¼ 0:

(23)

From Eqs. (22) and (23), a time-continuous state space model can be re-constructed as _ ¼ Ax þBu þNfðtÞ xðtÞ

(24)

yðtÞ ¼ Cx

(25)

S.Q. Zhang et al. / Journal of Sound and Vibration 333 (2014) 1209–1223

where u ¼ ϕa , y ¼ ϕs ,

( x¼

zr z_ r

1213

) ;

and f ¼ Fue denote the system input vector, the output vector, the state variable, and the unknown disturbance input. Additionally, A denotes the system matrix, B the voltage control matrix, N the disturbance influence matrix, and C the output matrix. They are given by " # " # 0 I 0 I 1 1 A¼ ¼ (26) 2 ~ C~ uu ~ K ~ M M  Ωr 2Λr Ωr uu uu uu " B¼

#

0

~ 1 ST Kuϕ M uu r "

N¼

0

(27)

#

~ 1 ST M uu r

C ¼ ½  K1 ϕϕ Kϕu Sr 0

(28)

(29)

in which Ωr and Λr are diagonal matrices composed by the first r eigen-frequencies and damping ratios, respectively. We define the control aim as zðtÞ ¼ FxðtÞ þ GuðtÞ

(30)

so that the controller should be designed to make zðtÞ as small as possible in the shortest possible amount of time. In vibration suppression, usually the desired output signal should be zero. Therefore the control aim is set to zðtÞ ¼ yðtÞ, which leads to F ¼ C and G ¼ 0. The state space model in Eqs. (24) and (25), which is obtained directly from the dynamic FE model of a smart structure, is called plant state space model in order to distinguish the extended state space model in later development. 3. Disturbance rejection control 3.1. Unknown disturbance estimation In the plant state space model given by Eqs. (24) and (25), the unknown disturbance input vector fðtÞ can be a force, a voltage or other kind of signals. The matrix N given in Eq. (28) is a force influence matrix, implying that an unknown force disturbance will be considered for exciting the structure. If a voltage disturbance is considered, the matrix N should equal the voltage control matrix B. Basically, any nonlinear disturbance can be exactly expressed by finite base functions vðtÞ as the linear part and residual error as fðtÞ ¼ HvðtÞ þ ΔðtÞ:

(31)

Here, ΔðtÞ is the residual error, which is assumed to be small such that it will be neglected in most cases. In such a way, the dynamics of the nonlinear unknown disturbance will be linearly approximated as fðtÞ  HvðtÞ

(32)

_ ¼ VvðtÞ: vðtÞ

(33)

A suitable choice of the matrices H and V requires usually a good understanding of the system behavior. One simple way to realize the linear formulation of the unknown disturbances is using Fourier series, which is an expansion of a periodic function fðtÞ in terms of an infinite sum of sines and cosines. So that the components of an unknown disturbance vector will be expressed as 1

f i  ai0 þ ∑ ðaij cos ðωij tÞ þ bij sin ðωij tÞÞ:

(34)

j¼1

The simplest choice for approximation of fi is retaining the constant terms only, which leads to the disturbance vector fðtÞ as a step function. If the observer is fast enough, it will follow the disturbance only by step functions [73]. As a result, H and V are derived as an identity matrix and a zero matrix, respectively, H ¼ I;

V¼0

(35)

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which leads to a PI observer [73]. Moreover, the ith disturbance can also be approximated by sine or cosine functions, for example, as f i  ai0 þ ai1 cos ðωi1 tÞ:

(36)

Here, ωij is the angular frequency of sine or cosine base functions, which can be given as any value depending on the signal. If the disturbances are periodic signals with the frequencies known, ωij can be given by the disturbance frequencies, which lead to a much better dynamic response than using only step function, especially when the structure is under periodic disturbances. If only one disturbance is considered, H and V can be derived as 2 3 0 0 0 6 0 0 ω 7 H ¼ ½1 1 0; V ¼ 4 (37) 5 0 ω 0 in which ω ¼ ω11 . The fictitious model of disturbances constructed based on Eq. (36) is nonlinearly expressed by cosine functions. However, it also can be constructed by other nonlinear functions, e.g. sine or polynomial ones, which will lead to H a I and V a 0 analogously. 3.2. Extended system Substituting the fictitious model of disturbances given in Eqs. (32) and (33) into the plant state space model in Eqs. (24) and (25) yields the extended state space model as       x B x_ ¼ Ae þ u (38) v 0 v_ y ¼ Ce

  x

(39)

v

with  Ae ¼

A

NH

0

V

 ;

Ce ¼ ½C 0:

According to the classical Luenberger observer structure, the extended observer system can be obtained as " # ( )  ( )   Lx A NH B x^ x^_ ^ ðy  yÞ ¼ þ u þ _ ^ L 0 V 0 v v v^ ( y^ ¼ ½C 0

x^ v^

(40)

(41)

) :

(42)

Here, the observer gains Lx and Lv can be calculated by using classical ways, like the pole placement design method, to make the extended model asymptotically stable if the extended system is detectable. Additionally, the number of measurements must be not less than the number of unknown disturbances [73]. From the extended observer model according to Eq. (41), the estimated state variables x^_ and v^_ can be expressed separately as ^ x^_ ¼ Ax^ þNHv^ þBu þLx ðy  yÞ

(43)

^ v^_ ¼ Vv^ þ Lv ðy  yÞ:

(44)

Solving the linear ordinary differential equation of the first order given in Eq. (44), we get v^ as Z t ^ ðexpðVðt  τÞÞLv ðyðτÞ  yðτÞÞÞ dτ þ expðVtÞv^ 0 v^ ¼

(45)

0

in which v^ 0 is the initial value of v^ at time t¼0, which is usually treated as zero, and expð□Þ is the exponential operator. Considering v^ 0 ¼ 0 and substituting Eq. (45) into Eq. (43) one obtains Z t ^ ^ ðexpðVðt  τÞÞLv ðyðτÞ  yðτÞÞÞ dτ þ Lx ðy  yÞ: (46) x^_ ¼ Ax^ þBu þ NH 0

Here, the estimating procedure for the state variable x by taking the proportional and weighted integral of measurement error y  y^ into account, which is a general expression of PI observer (H ¼ I and V ¼ 0) stated in [76,77], is defined as generalized proportional-integral (GPI) observer.

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3.3. Estimation error dynamic analysis In order to guarantee the stability and convergence of the observer system, the estimation error dynamic model has to be constructed and analyzed. In the smart structure system, the estimation error of the plant state variable and the disturbance are of importance, which are respectively defined as ^ ex ¼ x  x;

^ ef ¼ f Hv:

(47)

For simplicity, using the assumption of eliminating the residual error ΔðtÞ in Eq. (31), the estimation error of disturbance will be expressed as ^ þ Δ: ef ¼ Hðv  vÞ

(48)

Therefore, a new estimation error can be introduced to represent ef given as ^ ev ¼ v  v:

(49)

So the dynamic model of the estimation error for the extended observer system can be obtained as ( ) ( )   ex e_ x NΔ ¼ Ab þ ev e_ v 0 where the system matrix of the estimation error dynamic model Ab is " # Lx C : Ab ¼ Ae  Lv e

(50)

(51)

One of our control aims is to make the estimation error in Eq. (50) convergent to zero as fast as possible. The system matrix of error dynamics Ab determines the dynamic behavior of the error model, which is dependent not only on the system matrices of the plant but also on the observer gains. Due to the unchangeable properties of the plant matrices, the only way that can improve the error dynamics is to design the observer gains. 3.4. Riccati approach In order to estimate the state variables successfully, the error dynamics have to be stabilized and the estimation errors have to converge to zero as soon as possible. According to the Lyapunov stability criterion for the linear model, the estimation error model is asymptotically stable, if the Lyapunov algebraic equation is satisfied ATb P þ PAb ¼  Q

(52)

where Q is an arbitrary symmetric positive definite matrix. The estimation error is asymptotically stable if and only if for any Q ¼ Q T 4 0 there exists a unique P ¼ PT 4 0, such that the Lyapunov algebraic equation given by Eq. (52) is satisfied. Further, the observer gains are assumed to be calculated as ½LTx LTv  ¼ Ce P1 :

(53)

Substituting Eqs. (51) and (53) into Eq. (52) yields the algebraic Riccati equation as Ae P1 þP1 ATe  2P1 CTe Ce P1 þ P1 QP1 ¼ 0:

(54)

2

Since Q can be an arbitrary symmetric positive definite matrix, we define Q ¼ bP ðb 4 0Þ, so that a standard algebraic Riccati equation is obtained. Usually, a larger b produces larger observer gains, which shortens the rise time of the estimated signal but with larger overshoot. However, a large b will amplify the noises in the system as well, and the Riccati equation may become unsolvable if b is extremely large. 3.5. Closed-loop system As described before, the values of the extended observer will be fed back to the controller, by which the free vibration and the forced vibration will be respectively counteracted. Therefore, the control action should consist of two parts, one is ^ ^ Kx xðtÞ for counteracting the free vibration, and the other Kv vðtÞ for the forced vibration as ^  Kv vðtÞ ^ uðtÞ ¼  Kx xðtÞ

(55)

in which Kx can be designed with any ordinary method like pole placement, linear quadratic regulator, etc. In the later simulation, the control gain Kx is derived by linear quadratic regulator control method as T Kx ¼ R 1 r ðB Pþ Nr Þ:

(56)

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This gain makes the sum of the system input and output energy minimum. The total energy can be evaluated as Z 1 J lqr ¼ ðzðtÞT Q zðtÞ þuðtÞT RuðtÞÞ dt:

(57)

t0

Here, Q and R are the symmetric positive definite weighting matrices on the controlled output and control input, respectively, which can be approximated by Bryson's rule [80] as Q ii ¼

1 ; maxðjzi j2 Þ

R ii ¼

1 : maxðjui j2 Þ

(58)

In the control gain expression given by Eq. (56), P is a symmetric positive definite matrix, which is obtained by the following algebraic Riccati equation: T T AT P þPA þQ r  ðPB þ Nr ÞR 1 r ðB P þ Nr Þ ¼ 0

(59)

Rr ¼ GT Q G þR ;

(60)

with Q r ¼ FT Q F;

Nr ¼ FT Q G:

However, the control gain Kv , which compensates the unknown disturbance effects, can be obtained in a specific manner under the assumption of linear mapping X existing between x and v as x ¼ Xv

(61)

such that the time derivative of the state variable can be obtained as x_ ¼ Xv_ ¼ XVv:

(62)

Further substituting the linear mapping into the state space model of the plant yields the equilibrium equation for computation of the control gain Kv as ðA  BKx ÞX  XV BKv þNH ¼ 0

(63)

ðF  GKx ÞX GKv ¼ 0:

(64)

Since the unknown matrix X respectively appears on the right and left of the first two terms in Eq. (63), it is difficult to solve the above equations. Using the fictitious model given in Eq. (37) and assuming that the unknown matrices X and Kv are composed by three parts as X ¼ ½X1 X2 X3 

(65)

Kv ¼ ½Kv1 Kv2 Kv3 ;

(66)

six equilibrium equations can be obtained from Eqs. (63) and (64) as ðA  BKx ÞX1 BKv1 ¼ NH1

(67)

ðA  BKx ÞX2  ωX3  BKv2 ¼  NH2

(68)

ðA  BKx ÞX3 þ ωX2  BKv3 ¼  NH3

(69)

ðF  GKx ÞX1  GKv1 ¼ 0

(70)

ðF  GKx ÞX2  GKv2 ¼ 0

(71)

ðF  GKx ÞX3  GKv3 ¼ 0

(72)

where H ¼ ½H1 H2 H3 . Rearranging Eqs. (67)–(72) yields a linear equation in matrix form as 2 32 3 2 3 X1 A  BKx 0 0 B 0 0  NH1 6 76 7 6  ωI 0 B 0 76 X2 7 6  NH2 7 0 A BKx 6 7 6 76 7 6 7 6 6 7 0 0  B7 0 ωI A  BK  NH3 7 x 6 76 X3 7 6 6 7: 76 6 7¼6 0 0 G 0 0 76 Kv1 7 6 0 7 6 F GKx 7 6 76 7 6 7 6 6 7 0 0 G 0 7 0 F GKx 54 Kv2 5 4 0 5 4 Kv3 0 0 G 0 0 F  GKx 0

(73)

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Substituting Eq. (55) into Eq. (24) yields the closed-loop control system with the estimation of unknown disturbance using PI or GPI observer as 8 9 2 A  BKx > = < x_ > 6 0 e_ x ¼ 4 > ; : ^_ > 0 v

BKx A Lx C Lv C

38 9 2 3  BKv > N = 6 7  NH 7 5 ex þ 4 N 5f: > ; : ^ > V 0 v

(74)

Fig. 2. Cantilevered beam with a piezoelectric patch bonded.

Fig. 3. The dynamic behavior of the smart beam under a step disturbance force: (a) the sensor voltages, (b) the control voltages, (c) the estimated disturbances.

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4. Active control simulation A cantilevered beam with two collocated piezoelectric patches bonded on both surfaces at a distance of 50 mm from the cantilevered end, as shown in Fig. 2, is used for the simulation of vibration control. The dynamic FE model is derived by a mesh of 5  1 eight-node isoparametric shell elements along the Θ1 and Θ2 axes. The upper piezoelectric patch acts as a sensor and the lower one as an actuator. The patches have opposite polarization in the direction of the outward normal vectors of

Fig. 4. Estimated step signals using different b.

Fig. 5. The dynamic behavior of the smart beam under a harmonic disturbance force: (a) the sensor voltages, (b) the control voltages, (c) the estimated disturbances.

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the upper and lower surfaces. The dimensions of the master structure are 350  25  0.8 mm, and those of the piezoelectric patches are 75  25  0.25 mm. The master structure is made up of spring steel with Young's modulus E ¼ 210 GPa, Poisson's ratio ν ¼ 0:3, and the density ρ ¼ 7900 kg=m3 . The material properties of the piezoelectric patches are taken as E¼67 GPa, ν¼0.3, d31 ¼ d32 ¼  2:1  10  10 C/N, ϵ33 ¼ 2:13  10  8 F=m, and ρ ¼ 7800 kg=m3 . The damping matrix is obtained by the Rayleigh damping coefficient computation method with a damping ratio of 0.8 percent for the first six modes. Due to large size of the FE model, the system is reduced to the first 12 modes. The disturbances are assumed to be concentrated forces which are applied at the tip point. The simulations are carried out under various kinds of unknown disturbance forces, namely the step disturbance, the harmonic disturbance, the triangle wave disturbance, and the random disturbance. The control gain matrix Kx for DR control is derived based on the weighting matrices of Q ¼ 1=ð10Þ2 and R ¼ 1=ð200Þ2 (see Eq. (58)) through all the cases, which are also used in the simulations of LQR control. The observer gains are obtained by solving the algebraic Riccati equation, Eq. (54), using b ¼100 in all cases if it is not mentioned otherwise. 4.1. Step disturbance Firstly, the cantilevered smart beam is excited by a step disturbance force, which occurs at 0.5 s with the amplitude of 0.1 N. The vibration is suppressed by LQR control and DR control with PI and GPI observers, respectively, which is shown in Fig. 3. The uncontrolled and controlled sensor signals are displayed in Fig. 3(a), and the corresponding control voltages applied on the actuator are shown in Fig. 3(b). The figures illustrate that the DR control methods which take the unknown disturbance into account by using PI or GPI observer lead to better vibration suppression than LQR control. The estimated step disturbances by PI and GPI observers are displayed in Fig. 3(c), from which it can be seen that the signal estimated by the GPI observer has shorter rise time but larger overshoot than that estimated by the PI observer. As described above, the observer gains are affected by the parameter b in Eq. (54), which determines the observer dynamics. The estimated signals of the step disturbance by PI and GPI observers using different b, varying from b ¼ 1  10  3 to b ¼ 1  1010 , are calculated

Fig. 6. The dynamic behavior of the smart beam under a triangle wave disturbance force: (a) the sensor voltages, (b) the control voltages, (c) the estimated disturbances.

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and shown in Fig. 4. Generally, the signal estimated by the GPI observer rises much faster than that by the PI observer with the same b, but has larger overshoot than that estimated by the PI observer, implying that the GPI observer has better dynamic behavior. Interestingly, the rise time of the disturbance estimated by the PI observer decreases with b increasing from b ¼ 1  10  3 to about b¼1, and then the rise time increases, but slowly. A similar phenomenon occurs in the results derived by the GPI observer. 4.2. Harmonic disturbance In the second validation test, a harmonic disturbance force with an angular frequency of π rad/s is applied on the tip point of the smart beam. The harmonic disturbance follows the function fðtÞ ¼ 0:1  cos ðπtÞ N. The sensor signals and the control input signals are presented in Fig. 5(a) and (b), respectively. If the disturbance is absolutely unknown, ω¼1 rad/s is considered, otherwise ω ¼ π rad=s is employed if the disturbance frequency is known. The controlled sensor signals in Fig. 5(a) illustrate that the vibrations suppressed by DR control with PI or GPI observer have smaller amplitudes than those obtained by LQR control. Furthermore, the DR control with GPI observer (ω¼1 rad/s) has a better capability for vibration suppression than the controller with PI observer. If the angular frequency of the disturbance, i.e. ω ¼ π rad=s, is considered, the beam is perfectly damped as shown in Fig. 5(a). The estimations of the disturbance are displayed in Fig. 5(c), implying that the GPI observer with known angular frequency predicts the disturbance almost exactly. The GPI observer with unknown angular frequency and the PI observer estimate almost the same disturbances but with a slight time delay, which results in bigger sensor amplitudes than those of the GPI observer with known angular frequency in Fig. 5(a). Due to the better dynamic behavior of the GPI observer, even the angular frequency is unknown, the disturbance estimated by the GPI observer with ω ¼ 1 rad=s has a smaller time delay than that by the PI observer, which results in better vibration suppression. This can be explained by the fact that the first eigen-frequency of the beam is very high compared to the excitation frequency. Therefore even only a slight time delay can cause big differences concerning the vibration suppression effect.

Fig. 7. The dynamic behavior of the smart beam under a random disturbance force: (a) the sensor voltages, (b) the control voltages, (c) the estimated disturbances.

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4.3. Triangle wave disturbance A triangle periodic wave disturbance force with the angular frequency of ω ¼ π rad=s and the amplitude of 0.1 N is simulated in the third validation test. The controlled vibration and the control input are displayed in Fig. 6(a) and (b), respectively. Similar effects on vibration suppression by various control schemes can be observed as in the previous simulation. The vibration response is better suppressed by DR control with either PI or GPI observer than LQR control. The best result is obtained by the GPI observer especially if the angular frequency of the triangle wave disturbance is known. The estimated disturbances are displayed in Fig. 6(c), which illustrate that all the observers can estimate the disturbance very well. The GPI observer with the angular frequency of the disturbance known predicts the closest signal to the original disturbance among those three. 4.4. Random disturbance Finally, a random disturbance force is considered for the validation test. The sensor voltages and the control signals obtained by LQR control and DR control with PI and GPI observers are respectively displayed in Fig. 7(a) and (b). The time histories of the disturbance force and the estimated signals are shown in Fig. 7(c). DR control with either PI or GPI observer provides a significant damping. Again the best results are obtained by DR control with GPI observer. 5. Conclusion An electro-mechanically coupled dynamic FE model of smart structures is developed based on the FOSD hypothesis, in which an eight-node quadrilateral element with five mechanical degrees of freedom per node and one electrical degree of freedom per smart material layer is considered. Due to small strains and weak electric field, linear piezoelectric constitutive equations and the assumption of constant electric field through the thickness have been employed. Based on the dynamic FE model, the DR control with both PI and GPI observers has been developed for vibration suppression of smart structures. The unknown disturbances are estimated by PI or GPI observer, and the estimated signals are then fed back to the controller as measured signals. The proposed DR control has been implemented into the vibration control of a piezoelectric bonded cantilevered smart beam under various kinds of unknown disturbances. The damped dynamic response has been simulated and compared with each other, as well as to that derived by LQR control. The results illustrate that the vibrations have been better suppressed by DR control than those by LQR control. Among the results obtained by DR control with PI observer, GPI observer with the disturbance angular frequencies known and unknown, the vibrations have been most excellently controlled by using DR control with the frequencies known when the smart beam is under periodic disturbances. 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